Wireless Mobile Communication and Transmission Lab. Theory and Technology of Error Control Coding Chapter 7 Low Density Parity Check Codes

Wireless Mobile Communication and Transmission Lab. 2/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

Wireless Mobile Communication and Transmission Lab. 3/42 Introduction of LDPC codes Gallager Zyablov PinskerTanner MacKay Neal Wiberg Davey MacKay Yu Kou Shu Lin Fossorier SY Chung Urbanke Richardson Burshtein Miller McEliece Luby Mitzenmacher Spielman Some important research of LDPC codes since 1962

Wireless Mobile Communication and Transmission Lab. 4/42 Introduction of LDPC codes  Regular LDPC code(6,4)  parity check matrix H  Two classes of nodes in a Tanner  graph (variable nodes and check nodes)  Check node j is connected to variable  node i whenever element in H is 1  Bold line constructs a cycle  of length 6 in a Tanner Graph

Wireless Mobile Communication and Transmission Lab. 5/42 Introduction of LDPC codes

Wireless Mobile Communication and Transmission Lab. 6/42 Introduction of LDPC codes  rate=1/4, AWGN Channel, Thesis of M. C. Davey

Wireless Mobile Communication and Transmission Lab. 7/42 Introduction of LDPC codes  Local girth distribution histogram of variable nodes  Block length approaching infinity, the assumption of cycle freeness is asymptotically fulfilled  The relationship of girth, minimum distance and performance

Wireless Mobile Communication and Transmission Lab. 8/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

Wireless Mobile Communication and Transmission Lab. 9/42 Encoding of LDPC codes  H=[P|I]  G ＝ [I|P’]  C=M*G

Wireless Mobile Communication and Transmission Lab. 10/42 Encoding of LDPC codes

Wireless Mobile Communication and Transmission Lab. 11/42 Encoding of LDPC codes

Wireless Mobile Communication and Transmission Lab. 12/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

Wireless Mobile Communication and Transmission Lab. 13/42 Construction of parity check matrix  Random construction methods  Structured construction methods

Wireless Mobile Communication and Transmission Lab. 14/42 Construction of parity check matrix  Gallager method

Wireless Mobile Communication and Transmission Lab. 15/42 Construction of parity check matrix  Mackay methods

Wireless Mobile Communication and Transmission Lab. 16/42 Construction of parity check matrix  Bit-filling

Wireless Mobile Communication and Transmission Lab. 17/42 Construction of parity check matrix  Extended Bit-filling

Wireless Mobile Communication and Transmission Lab. 18/42 Construction of parity check matrix  Hesuristic girth distribution

Wireless Mobile Communication and Transmission Lab. 19/42 Construction of parity check matrix  Progressive edge growth (PEG)

Wireless Mobile Communication and Transmission Lab. 20/42 Construction of parity check matrix  Random construction methods  Structured construction methods

Wireless Mobile Communication and Transmission Lab. 21/42 Construction of parity check matrix  FG-LDPC:EG-LDPC and PG-LDPC  n points and J lines : n*J incidense matrix H  Each line is composed of p points  There is one and only one line between two points  Each point lies on q lines  Any pare of lines has only one common point or no common point

Wireless Mobile Communication and Transmission Lab. 22/42 Construction of parity check matrix  Partial geometry LDPC Steiner 2-design; Net or transversal design (TD); Generalized quadrangle (GQ); Proper PG

Wireless Mobile Communication and Transmission Lab. 23/42 Construction of parity check matrix  BIBD-LDPC

Wireless Mobile Communication and Transmission Lab. 24/42 Construction of parity check matrix  Block-LDPC

Wireless Mobile Communication and Transmission Lab. 25/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

Wireless Mobile Communication and Transmission Lab. 26/42 Decoding of LDPC codes  Bit flipping method  Belief propagation and related methods  Weighted bit flipping methods

Wireless Mobile Communication and Transmission Lab. 27/42 Decoding of LDPC codes  Bit flipping method =0 =1 Connected to two unsatisfied check nodes: flipped

Wireless Mobile Communication and Transmission Lab. 28/42 Decoding of LDPC codes  Bit flipping method  Belief propagation and related methods  Weighted bit flipping methods

Wireless Mobile Communication and Transmission Lab. 29/42 Decoding of LDPC codes  Belief propagation method All the effective decoding strategies for LDPC codes are message passing algorithms The best algorithm known is the Belief Propagation algorithm (1) Complicated calculations are distributed among simple node processors (2) After several iterations, the solution of the global problem is available (3) BP algorithm is the optimal if there are no cycles or ignore cycles

Wireless Mobile Communication and Transmission Lab. 30/42 Decoding of LDPC codes  Belief propagation method (log domain) Probability information transmitting among connected codes through the edge Two types of message: The probability that one bit is 1 or 0, obtained via the connected checks nodes other than the check node that received the probability. The conditional probability of that one check node is satisfied if one connected bit is 1 or 0

Wireless Mobile Communication and Transmission Lab. 31/42 Decoding of LDPC codes  Belief propagation method: message passing in two steps

Wireless Mobile Communication and Transmission Lab. 32/42 Decoding of LDPC codes  UMP-BP based (min sum)

Wireless Mobile Communication and Transmission Lab. 33/42 Decoding of LDPC codes  Normalized UMP-BP based Reduce the complexity of horizontal step: The function value is greatly decided by the variable with minimum absolute value, L2 is greater than L1, Normalized factor is used to compensate the performance loss

Wireless Mobile Communication and Transmission Lab. 34/42 Decoding of LDPC codes  Bit flipping method  Belief propagation and related methods  Weighted bit flipping methods

Wireless Mobile Communication and Transmission Lab. 35/42 BPSK Modulation: The smaller the absolute value, the fewer the reliability Output of the check node Flipping the variable node n with largest weight Decoding of LDPC codes  Weighted bit flipping methods

Wireless Mobile Communication and Transmission Lab. 36/42 Decoding of LDPC codes Some improvements of WBF algorithm Consider the reliability of the bit (MWBF): Modified check node output (IMWBF):  Weighted bit flipping methods

Wireless Mobile Communication and Transmission Lab. 37/42 Decoding of LDPC codes Some improvements of WBF algorithm Consider both of the maximum and minimum symbols (LP): Add a check weight factor (MLP): Consider the ratio (RRWBF):  Weighted bit flipping methods

Wireless Mobile Communication and Transmission Lab. 38/42 Decoding of LDPC codes Developed from IMWBF which is a counterpart to Normalized BP Based algorithm Consider all the symbol in each check with the constraint of extrinsic information: Linear combination  Weighted bit flipping methods

Wireless Mobile Communication and Transmission Lab. 39/42 Outline  Introduction of LDPC codes  Encoding of LDPC codes  Construction of parity check matrix  Decoding of LDPC codes  Density evolution and EXIT

Wireless Mobile Communication and Transmission Lab. 40/42 Density Evolution  Messages passed in the factor graph are random variables. The calculations performed under the SPA are functions of random variables.  Messages passed through the graph are conditionally independent  Symmetry Condition

Wireless Mobile Communication and Transmission Lab. 41/42 EXIT VNDCND AWGN channel output Iterative Decoding of LDPC Decision

Wireless Mobile Communication and Transmission Lab. 42/42 EXIT